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Mathematics > Differential Geometry

Abstract: A comparison theorem for the isoperimetric profile on the universal cover of
surfaces evolving by normalised Ricci flow is proven. For any initial metric, a
model comparison is constructed that initially lies below the profile of the
initial metric and which converges to the profile of the constant curvature
metric. The comparison theorem implies that the evolving metric is bounded
below by the model comparison for all time and hence converges to the constant
curvature profile. This yields a curvature bound and a bound on the
isoperimetric constant, leading to a direct proof that the metric converges to
the constant curvature metric.